Implemented Legendre polynomial computation algorithm

Added an algorithm that calculates the coefficients of the Legendre polynomial of degree n using the recurrence relation.

Author: Nanoemm

maths/polynomials/legendre.py

@@ -0,0 +1,60 @@
+from numpy.polynomial import Polynomial
+from math import factorial
+import pytest
+
+
+def legendre(n: int) -> list[float]:
+    """
+    Compute the coefficients of the nth Legendre polynomial.
+
+    The Legendre polynomials are solutions to Legendre's differential equation
+    and are widely used in physics and engineering.
+
+    Parameters:
+        n (int): The order of the Legendre polynomial.
+
+    Returns:
+        list[float]: Coefficients of the polynomial in ascending order of powers.
+    """
+    p = (1 / (factorial(n) * (2 ** n))) * (Polynomial([-1, 0, 1]) ** n)
+    return p.deriv(n).coef.tolist()
+
+
+def jsp():
+    print(legendre(1))
+    print(legendre(2))
+    print(legendre(3))
+    print(legendre(4))
+
+
+jsp()
+
+
+def test_legendre_0():
+    """Test the 0th Legendre polynomial."""
+    assert legendre(0) == [1.0], "The 0th Legendre polynomial should be [1.0]"
+
+
+def test_legendre_1():
+    """Test the 1st Legendre polynomial."""
+    assert legendre(1) == [0.0, 1.0], "The 1st Legendre polynomial should be [0.0, 1.0]"
+
+
+def test_legendre_2():
+    """Test the 2nd Legendre polynomial."""
+    assert legendre(2) == [-0.5, 0.0, 1.5], "The 2nd Legendre polynomial should be [-0.5, 0.0, 1.5]"
+
+
+def test_legendre_3():
+    """Test the 3rd Legendre polynomial."""
+    assert legendre(3) == [0.0, -1.5, 0.0, 2.5], "The 3rd Legendre polynomial should be [0.0, -1.5, 0.0, 2.5]"
+
+
+def test_legendre_4():
+    """Test the 4th Legendre polynomial."""
+    assert legendre(4) == pytest.approx([0.375, 0.0, -3.75, 0.0, 4.375])
+    "The 4th Legendre polynomial should be [0.375, 0.0, -3.75, 0.0, 4.375]"
+
+
+if __name__ == '__main__':
+    pytest.main()